# Cognitively Economical Heuristic for Multiple Sequence Alignment under Uncertainties

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## Abstract

**:**

## 1. Introduction

- m is a nonnegative integer ($m\in {\mathbb{N}}_{0}$),
- sequence ${s}_{i}$ represents recognized objects, i.e.,$$(\forall \phantom{\rule{0.277778em}{0ex}}0\le k<m)({s}_{i}\left[k\right]\in S)\phantom{\rule{0.277778em}{0ex}},$$
- sequence ${c}_{i}$ contains the corresponding real-valued recognition confidences for objects in ${s}_{i}$.

## 2. Background and Related Work

- $d\left({s}_{i}[k-1]\right)$ is the cost of deletion of symbol ${s}_{i}[k-1]$,
- $i\left({s}_{j}[l-1]\right)$ is the cost of insertion of symbol ${s}_{j}[l-1]$,
- $r({s}_{i}[k-1],{s}_{j}[l-1])$ is the cost of replacement of symbol ${s}_{i}[k-1]$ by symbol ${s}_{j}[l-1]$.

- (i)
- The number of clusters is determined prior to the pairwise sequence alignment. Each distinct maximum-length sequence in a set of recognition hypotheses is declared as a cluster representative. All other non-maximum-length sequences were then assigned to the closest cluster representative, where the distance between two sequences was calculated by means of the adapted edit distance algorithm.
- (ii)
- The proposed adaptation of the edit distance approach is inspired by human working memory limitations (cf. [26]). To reduce the “cognitive load” of our approach, we consider only “economical” sequence alignments that are optimal in terms of the standard minimum edit distance approach and in which no space is inserted into longer sequence. These two requirements for cognitive economy allow for the substantial reduction of the number of sequences derived in the alignment process by means of padding (as detailed in Section 3).

## 3. Heuristic for Multiple Sequence Alignment

- The gap-minimum alignment algorithm, introduced and illustrated in Section 3.1, is intended for alignment of two recognition hypotheses.
- The cluster-based voting algorithm, introduced and illustrated in Section 3.2, builds upon the first algorithm and is intended for multiple recognition hypothesis alignment, based on which a single recognition result is derived.

#### 3.1. Gap-Minimum Two Sequence Alignment

**Step 1.1:**If sequences ${s}_{i}$ and ${s}_{j}$ are of equal length, i.e., $m=n$, then no particular alignment is performed, i.e.:

**Step 1.2:**Otherwise, if sequences ${s}_{i}$ and ${s}_{j}$ are not of equal length, let us assume, without loss of generality, that the length of ${s}_{i}$ is less than the length of ${s}_{j}$, i.e., $m<n$. A distance matrix is generated, with the costs of all edit operations set to one, as described in Section 2. Let P be a set of all optimal alignments of sequences ${s}_{i}$ and ${s}_{j}$ derived from the distance matrix. We recall that all alignments in set P are determined by means of the minimal edit distance algorithm (i.e., the Levenshtein algorithm), and thus they contain a minimal number of single-symbol edit operations (i.e., deletion, insertion, and replacement) required to transform sequence ${s}_{i}$ into sequence ${s}_{j}$. In general, it is easy to show that set P is never empty (i.e., it is always possible to find at least one alignment).

**Step 1.3:**From set P, containing all optimal alignments of sequences ${s}_{i}$ and ${s}_{j}$, we select only those alignments in which no space is inserted into longer a sequence. Let ${P}_{r}\subseteq P$ be a set of selected alignments. If ${P}_{r}\ne \varnothing $, it is declared that sequences ${s}_{i}$ and ${s}_{j}$ cannot be economically aligned, and the alignment process is terminated. Otherwise, the algorithm proceeds to the next step.

**Step 1.4:**The value of each alignment $p\in {P}_{r}$ is calculated as the sum of confidence values of all symbols in a longer sequence ${s}_{j}$ that are opposed to a space, i.e.:

**Example**

**1.**

**Example**

**2.**

#### 3.2. Cluster-Based Multiple Sequence Alignment

**Step 2.1:**Let ${H}_{t}$ be a multiset containing recognition hypotheses from H with the maximum length, i.e.,

**Step 2.2:**If $|{H}_{t}|<|H|$, each recognition hypothesis from set $H\backslash {H}_{t}$ is either assigned to exactly one cluster or discarded. More particularly, each hypothesis $h\in H\backslash {H}_{t}$ is independently aligned—by means of cognitively economical gap-minimum sequence alignment introduced in Section 3.1—to all hypotheses from set ${H}_{t}$, producing a set of alignments:

**Step 2.3:**It is easy to show that all recognition hypotheses (some of them being transformed by adding spaces) assigned to the clusters are of equal length ${L}_{max}=\underset{(s,c)\in H}{max}\phantom{\rule{0.277778em}{0ex}}\left|s\right|$. In this step, they are all arrayed in rows, each of which contains ${L}_{max}$ columns, and the order or rows is irrelevant. A new sequence ${s}_{f}$ containing ${L}_{max}$ symbols—one for each column—is generated by means of voting. For each column, a symbol from set $S\cup \{\u25b3\}$ with the maximum sum of confidence values in the given column is selected. The final recognition result is obtained by removing all spaces from ${s}_{f}$.

**Example**

**3.**

## 4. Evaluation and Discussion

#### 4.1. Extrinsic Evaluation

- Hardware and software: the subjects used Android-based mobile phones of the same type and with the same software settings.
- Subjects: the subjects were of the same gender (male), and comparable in height and expertise in recording electric meters with an Android-based mobile phone. They did not have any insight into the post-processing results.
- Electricity meters: both subjects recorded the same set of 100 electricity meters, including 5 m with one rate, and 95 m with two rates.
- Ambient: to achieve the same ambient conditions, each electricity meter was recorded first by one subject and then immediately after by another.
- Recording span: when reading an electricity meter, the digit recognition system was set to record until ten image frames were recorded or the recording time reached three seconds.

- A total of 86 of 384 recognition hypothesis sets do not contain correct recognition hypotheses (39 sets in Corpus 1 and 47 sets in Corpus 2). However, the hypotheses from seven of these sets were correctly aligned; i.e., the correct recognition results were derived (in 1 of 39 sets in Corpus 1 and 6 of 47 sets in Corpus 2). One of these sets and the derivation of the recognition result are presented above in Example 3.
- A total of 114 of 384 sets contain at least one correct recognition hypothesis, but the number of correct hypotheses in each of these sets is less than or equal to the half of the number of hypotheses in a given set (59 sets in Corpus 1 and 55 sets in Corpus 2). The correct recognition results were derived for 72 of these sets (40 of 59 sets in Corpus 1 and 32 of 55 sets in Corpus 2).
- A total 184 of 384 sets contain correct recognition hypotheses, and the number of correct hypotheses in each of these sets is greater than the half of the number of hypotheses in a given set (96 sets in Corpus 1 and 88 sets in Corpus 2). The correct recognition results were derived for all these sets except one (from Corpus 2).

# Correct Recognition | Corpus 1 | Corpus 2 | Total | |||
---|---|---|---|---|---|---|

Hypotheses in a Set | # Sets | # Correctly Aligned | # Sets | # Correctly Aligned | # Sets | # Correctly Aligned |

0 | 39 | 1 | 47 | 6 | 86 | 7 |

≤half of the set, ≠0 | 59 | 40 | 55 | 32 | 114 | 72 |

>half of the set | 96 | 96 | 88 | 87 | 184 | 183 |

Total | 194 | 137 | 190 | 125 | 384 | 262 |

#### 4.2. Comparison to Human Performance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A set of image frame segments derived from a video stream captured by a mobile phone application. The size of fragments is $230\times 52$ pixels (with 300 pixels/inch). For the purpose of presentation, images are scaled up.

**Figure 2.**Illustration of the minimum edit distance algorithm. The costs of all single-symbol edit operations are equal to one, cf. Equation (8). Abbreviations: i—insertion, d—deletion, r—replacement.

**Table 1.**For each frame given in Figure 1, a recognition hypothesis is provided. The digit after the decimal point is intentionally discarded. The recognition confidences are normalized to the range $[0,1]$.

Frame Recognition Hypotheses |
---|

${h}_{1}\equiv ({s}_{1},{c}_{1})\equiv (4,0.931),(7,0.834),(7,0.877)$ |

${h}_{2}\equiv ({s}_{2},{c}_{2})\equiv (4,0.933),(6,0.883),(7,0.828),(7,0.827)$ |

${h}_{3}\equiv ({s}_{3},{c}_{3})\equiv (4,0.907),(6,0.880),(7,0.829),(7,0.840)$ |

${h}_{4}\equiv ({s}_{4},{c}_{4})\equiv (4,0.928),(6,0.875),(7,0.843),(7,0.886)$ |

${h}_{5}\equiv ({s}_{5},{c}_{5})\equiv (4,0.921),(6,0.883),(7,0.851),(7,0.791),(8,0.640),(7,0.781)$ |

${h}_{6}\equiv ({s}_{6},{c}_{6})\equiv (4,0.909),(6,0.869),(7,0.836),(7,0.830)$ |

${h}_{7}\equiv ({s}_{7},{c}_{7})\equiv (4,0.907),(6,0.861),(7,0.833),(7,0.882)$ |

${h}_{8}\equiv ({s}_{8},{c}_{8})\equiv (4,0.881),(8,0.838),(7,0.809),(7,0.846),(8,0.651),(7,0.819)$ |

${h}_{9}\equiv ({s}_{9},{c}_{9})\equiv (4,0.891),(7,0.813),(7,0.860)$ |

**Table 2.**Illustration of the cluster-based multiple-sequence alignment algorithm. The recognition confidence values of the digits are given in Table 1. The recognition confidence value of a space is set to $0.5$.

Step 2.1 | Step 2.2 | Step 2.3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Hypotheses | Cluster | Hypotheses | Cluster | Arraying | |||||||

${h}_{1}$ | 477 | ? | ${\widehat{h}}_{1}$ | $4\u25b3\u25b37\u25b37$ | ${C}_{8}$ | 4 | △ | △ | 7 | △ | 7 |

${h}_{2}$ | 4677 | ? | ${\widehat{h}}_{2}$ | $4677\u25b3\u25b3$ | ${C}_{5}$ | 4 | 6 | 7 | 7 | △ | △ |

${h}_{3}$ | 4677 | ? | ${\widehat{h}}_{3}$ | $4677\u25b3\u25b3$ | ${C}_{5}$ | 4 | 6 | 7 | 7 | △ | △ |

${h}_{4}$ | 4677 | ? | ${\widehat{h}}_{4}$ | $4677\u25b3\u25b3$ | ${C}_{5}$ | 4 | 6 | 7 | 7 | △ | △ |

${h}_{5}$ | 467787 | ${C}_{5}$ | ${\widehat{h}}_{5}$ | 467787 | ${C}_{5}$ | 4 | 6 | 7 | 7 | 8 | 7 |

${h}_{6}$ | 4677 | ? | ${\widehat{h}}_{6}$ | $4677\u25b3\u25b3$ | ${C}_{5}$ | 4 | 6 | 7 | 7 | △ | △ |

${h}_{7}$ | 4677 | ? | ${\widehat{h}}_{7}$ | $4677\u25b3\u25b3$ | ${C}_{5}$ | 4 | 6 | 7 | 7 | △ | △ |

${h}_{8}$ | 487787 | ${C}_{8}$ | ${\widehat{h}}_{8}$ | 487787 | ${C}_{8}$ | 4 | 8 | 7 | 7 | 8 | 7 |

${h}_{9}$ | 477 | ? | ${\widehat{h}}_{9}$ | $4\u25b3\u25b37\u25b37$ | ${C}_{8}$ | 4 | △ | △ | 7 | △ | 7 |

↓ | ↓ | ↓ | ↓ | ↓ | ↓ | ||||||

Voting: | 4 | 6 | 7 | 7 | △ | 7 | |||||

Final result: | 46777 |

System | Image Frames | Corpus 1 | Corpus 2 | Total |
---|---|---|---|---|

External number | # image frames | 2011 | 1873 | 3884 |

recognition system [1] | average root mean square contrast | $0.227\phantom{\rule{0.277778em}{0ex}}(\pm 0.040)$ | $0.208\phantom{\rule{0.277778em}{0ex}}(\pm 0.039)$ | $0.218\phantom{\rule{0.277778em}{0ex}}(\pm 0.041)$ |

# correctly recognized image frames | $1012\phantom{\rule{0.277778em}{0ex}}(50.32\%)$ | $858\phantom{\rule{0.277778em}{0ex}}(45.81\%)$ | $1870\phantom{\rule{0.277778em}{0ex}}(48.15\%)$ | |

Proposed post-processing | # hypothesis sets | 194 | 190 | 384 |

subsystem | average # hypotheses per set | $10.366\phantom{\rule{0.277778em}{0ex}}(\pm 2.109)$ | $9.858\phantom{\rule{0.277778em}{0ex}}(\pm 1.538)$ | $10.116\phantom{\rule{0.277778em}{0ex}}(\pm 1.866)$ |

average# correct hypotheses per set | $5.201\phantom{\rule{0.277778em}{0ex}}(\pm 3.766)$ | $4.437\phantom{\rule{0.277778em}{0ex}}(\pm 3.647)$ | $4.823\phantom{\rule{0.277778em}{0ex}}(\pm 3.727)$ | |

# correctly aligned recognition result | $137\phantom{\rule{0.277778em}{0ex}}(70.62\%)$ | $125\phantom{\rule{0.277778em}{0ex}}(65.79\%)$ | $262\phantom{\rule{0.277778em}{0ex}}(68.23\%)$ |

**Table 5.**Confusion matrix for Corpus 1 (INS—segment incorrectly recognized as a digit; ND—digit not detected).

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ND | Total | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 92 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 3 | 99 |

1 | 0 | 90 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 93 |

2 | 0 | 0 | 94 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 6 | 101 |

3 | 0 | 2 | 0 | 115 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 121 |

4 | 0 | 1 | 0 | 0 | 101 | 0 | 0 | 0 | 0 | 0 | 3 | 105 |

5 | 0 | 1 | 0 | 1 | 0 | 74 | 1 | 0 | 0 | 0 | 1 | 78 |

6 | 4 | 0 | 0 | 1 | 0 | 1 | 80 | 0 | 2 | 0 | 6 | 94 |

7 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 86 | 4 | 0 | 4 | 95 |

8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 89 | 0 | 4 | 95 |

9 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 85 | 2 | 88 |

INS | 0 | 5 | 0 | 1 | 1 | 0 | 1 | 1 | 5 | 0 | – | 14 |

**Table 6.**Confusion matrix for Corpus 2 (INS—segment incorrectly recognized as a digit; ND—digit not detected).

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ND | Total | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 88 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 4 | 98 |

1 | 0 | 92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 93 |

2 | 0 | 0 | 89 | 0 | 4 | 0 | 0 | 4 | 0 | 0 | 2 | 99 |

3 | 0 | 0 | 0 | 103 | 0 | 0 | 0 | 0 | 0 | 1 | 6 | 110 |

4 | 0 | 0 | 0 | 0 | 99 | 0 | 0 | 1 | 0 | 0 | 5 | 105 |

5 | 1 | 0 | 0 | 4 | 0 | 70 | 0 | 0 | 0 | 0 | 1 | 76 |

6 | 6 | 0 | 0 | 1 | 0 | 3 | 78 | 0 | 1 | 0 | 4 | 93 |

7 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 88 | 1 | 0 | 1 | 92 |

8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 93 | 0 | 3 | 97 |

9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 80 | 2 | 86 |

INS | 0 | 12 | 1 | 5 | 1 | 0 | 2 | 2 | 10 | 9 | – | 42 |

# Sets | # Correctly Derived Results | Total Overlap | |||
---|---|---|---|---|---|

Heuristic | Human | Overlap | |||

C1 | $194\phantom{\rule{0.277778em}{0ex}}(100\%)$ | $137\phantom{\rule{0.277778em}{0ex}}(70.62\%)$ | $137\phantom{\rule{0.277778em}{0ex}}(70.62\%)$ | $129\phantom{\rule{0.277778em}{0ex}}(66.49\%)$ | $157\phantom{\rule{0.277778em}{0ex}}(80.93\%)$ |

C2 | $190\phantom{\rule{0.277778em}{0ex}}(100\%)$ | $125\phantom{\rule{0.277778em}{0ex}}(65.79\%)$ | $121\phantom{\rule{0.277778em}{0ex}}(63.68\%)$ | $114\phantom{\rule{0.277778em}{0ex}}(60\%)$ | $141\phantom{\rule{0.277778em}{0ex}}(74.21\%)$ |

Total | $384\phantom{\rule{0.277778em}{0ex}}(100\%)$ | $262\phantom{\rule{0.277778em}{0ex}}(68.23\%)$ | $258\phantom{\rule{0.277778em}{0ex}}(67.19\%)$ | $243\phantom{\rule{0.277778em}{0ex}}(63.28\%)$ | $298\phantom{\rule{0.277778em}{0ex}}(77.60\%)$ |

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## Share and Cite

**MDPI and ACS Style**

Gnjatović, M.; Maček, N.; Saračević, M.; Adamović, S.; Joksimović, D.; Karabašević, D.
Cognitively Economical Heuristic for Multiple Sequence Alignment under Uncertainties. *Axioms* **2023**, *12*, 3.
https://doi.org/10.3390/axioms12010003

**AMA Style**

Gnjatović M, Maček N, Saračević M, Adamović S, Joksimović D, Karabašević D.
Cognitively Economical Heuristic for Multiple Sequence Alignment under Uncertainties. *Axioms*. 2023; 12(1):3.
https://doi.org/10.3390/axioms12010003

**Chicago/Turabian Style**

Gnjatović, Milan, Nemanja Maček, Muzafer Saračević, Saša Adamović, Dušan Joksimović, and Darjan Karabašević.
2023. "Cognitively Economical Heuristic for Multiple Sequence Alignment under Uncertainties" *Axioms* 12, no. 1: 3.
https://doi.org/10.3390/axioms12010003